9-12 math i student edition unit 1 function families

8/9/2019 9-12 Math I Student Edition Unit 1 Function Families

1/61

Mathematics I Frameworks

Student Edition

Unit 1Function Families

2nd EditionMay 14, 2008

Georgia Department of Education


2/61

One Stop Shop For Educators

Mathematics I Unit 1 2nd Edition

Georgia Department of EducationKathy Cox, State Superintendent of Schools

May14, 2008Copyright 2008 All Rights Reserved

Unit 1: Page2 of61

Table of Contents

INTRODUCTION: ............................................................................................................................ 3Exploring Functions with Fiona Learning Task ................................................................................ 6Fences and Functions Learning Task............................................................................................... 15From Wonderland to Functionland Learning Task.......................................................................... 22Sequences as Functions Learning Task ........................................................................................... 36Walking, Falling, and Making Money ............................................................................................. 41Painted Cubes Learning Task .......................................................................................................... 55
http://www.georgiastandards.org/


3/61





Unit 1: Page3 of61

Mathematics 1Unit 1

Function Families

Student Edition

INTRODUCTION:

In seventh and eighth grade, students learned about functions generally and about linear functionsspecifically. This unit explores properties of basic quadratic, cubic, absolute value, square root,and rational functions and new language and notation for talking about functions. The discussionof function characteristics includes further development of the language of mathematical reasoningto include formal discussion of the logical relationships between an implication and its converse,inverse, and contrapositive.

ENDURING UNDERSTANDINGS:

Functions have three parts: (i) a domain, which is the set of inputs to the function, (ii) a

range, which is the set of outputs, and (iii) some rule or statement of correspond indicatinghow each input determines a unique output.

The domain and rule of correspondence determine the range.

Graphs are geometric representations of functions.

Functions are equal if and only if they have the same domain and rule of correspondence.

Function notation provides an efficient way to talk about functions, but notation is just that,an efficient way to talk about functions. The variables used to represent domain values,range values, and the function as a whole are arbitrary. Changing variable names does notchange the function.

Logical equivalence is a concept that applies to the form of a conditional statement. Everyconditional statement and its contrapositive are logically equivalent. Given a trueconditional statement, whether the converse or inverse of the conditional statement is alsotrue depends on the content of the statement. The converse and inverse forms are notlogically equivalent to the original conditional form.

KEY STANDARDS ADDRESSED:

MM1A1. Students will explore and interpret the characteristics of functions, using graphs,

tables, and simple algebraic techniques.

a. Represent functions using function notation.

b. Graph the basic functionsf(x) =xn where n = 1 to 3,f(x) = x ,f(x) = x , andf(x) = 1/x.

c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks, aswell as reflections across thex- andy-axes. [Previewed in this unit.]

d. Investigate and explain the characteristics of a function: domain, range, zeros, intercepts,intervals of increase and decrease, maximum and minimum values, and end behavior.
http://www.georgiastandards.org/


4/61





Unit 1: Page4 of61

e. Relate to a given context the characteristics of a function, and use graphs and tables toinvestigate its behavior.

f. Recognize sequences as functions with domains that are whole numbers.

g.

Explore rates of change, comparing constant rates of change (i.e., slope) versus variablerates of change. Compare rates of change of linear, quadratic, square root, and otherfunction families.

MM1G2. Students will understand and use the language of mathematical argument

and justification.

a. Use conjecture, inductive reasoning, deductive reasoning, counterexamples, and indirectproof as appropriate.

b. Understand and use the relationships among a statement and its converse, inverse, andcontrapositive.

Unit Overview:

Prior to this unit students need to have worked extensively with operations on integers,rational numbers, and square roots of nonnegative integers as indicated in the Grade 6 8standards for Number and Operations.

In the unit students will apply and extend the Grade 78 standards related to writingalgebraic expressions, evaluating quantities using algebraic expressions, understanding inequalitiesin one variable, and understanding relations and linear functions as they develop much deeper andmore sophisticated understanding of relationships between two variables. Students are assumed tohave a deep understanding of linear relationships between variable quantities. Students should

understand how to find the areas of triangles, rectangles, squares, and circles and the volumes ofrectangular solids.

The unit begins with intensive work with function notation. Students learn to use functionnotation to ask and answer questions about functional relationships presented tabular, graphical,and algebraic form. The distinction between discrete and continuous domains is explored throughcomparing and contrasting functions which have the same rule of correspondence, but differentdomains. Through extensive work with reading and drawing graphs, students learn to view graphsof functional relationships as whole objects rather than collections of individual points and toapply standard techniques used to draw representative graphs of functions with unboundeddomains.

The unit includes an introduction to propositional logic of conditional statements, theirconverses, inverses, and contrapositives. Conditional statements about the absolute value functionand vertical translations of the absolute value function give students concrete examples that can beclassified as true or false by examining graphs. Combining analysis of conditional statements withfurther exploration of basic functions demonstrates that conditional statements are importantthroughout the study of mathematics, not just in geometry where this material has traditionallybeen introduced.


5/61





Unit 1: Page5 of61

Average rate of change of a function is introduced through the interpretation of averagespeed, but is extended to the context of rate of change of revenue for varying quantities of aproduct sold. Students contrast constant rates of change to variable ones through the concept of

average rate of change. The basic function families are introduced primarily through real-worldcontexts that are modeled by these functions so that students understand that these functions arestudied because they are important for interpreting the world in which we live. The work withvertical shifts, stretches, and shrinks and reflections of graphs of basic functions is integratedthroughout the unit rather than studied as a topic in isolation. Functions whose graphs are relatedby one of these transformations arise from the context and the context promotes understanding ofthe relationship between a change in a formula and the corresponding change in the graph.

Students need facility in working with functions given via tables, graphs, or algebraicformulas, using function notation correctly, and learning to view a function as an entity to beanalyzed and compared to other functions.

Throughout this unit, it is important to:Begin exploration of a new function by generating a table of values using a variety of numbersfrom the domain. Decide, based on the context, what kinds of numbers can be in the domain,and make sure you choose negative numbers or numbers expressed as fractions or decimals ifsuch numbers are included in the domain.

Do extensive graphing by hand. Once students have a deep understanding of the relationshipsbetween formulas and graphs, regular use of graphing technology will be important. For thisintroductory unit, graphing by hand is necessary to develop understanding.

Be extremely careful in the use of language. Always use the name of the function, for examplef, to refer to the function as a whole and usef(x) to refer to the input when the input isx. Forexample, when language is used correctly, a graph of the functionfin the x, y-plane is the graph

of the equationy =f(x) since we graph those points, and only those points, of the form (x,y)where they-coordinates is equal tof(x).

TASKS: The remaining content of this framework consists of student learning tasks designed toallow students to learn by investigating situations with a real-world context. The first leaning taskis intended to launch the unit. Its primary focus is introducing function notation and the moreformal approach to functions characteristic of high school mathematics. The second throughseventh learning tasks extend students knowledge of functions through in depth consideration ofdomain, range, average rate of change, and other characteristics of functions basic to the study ofhigh school mathematics. The last task is designed to demonstrate the type of assessment activitiesstudents should be comfortable with by the end of the unit.


6/61





Unit 1: Page6 of61

Exploring Functions with Fiona Learning Task

1. While visiting her grandmother, Fiona Evans found markings on the inside of a closet door

showing the heights of her mother, Julia, and her mothers brothers and sisters on theirbirthdays growing up. From the markings in the closet, Fiona wrote down her mothers heighteach year from ages 2 to 16. Her grandmother found the measurements at birth and one yearby looking in her mothers baby book. The data is provided in the table below, with heightsrounded to the nearest inch.

Age (yrs.) x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Height (in.) y 21 30 35 39 43 46 48 51 53 55 59 62 64 65 65 66 66

a. Which variable is the independent variable, and which is the dependent variable? Explain

your choice.

b. Make a graph of the data.

c. Should you connect the dots on your graph? Explain.

d. Describe how Julias height changed as she grew up.

e. How tall was Julia on her 11th birthday? Explain how you can see this in both the graphand the table.

f. What do you think happened to Julias height after age 16? Explain. How could you showthis on your graph?


7/61





Unit 1: Page7 of61

In Mathematics I and all advanced mathematics,function notation is used as an efficient way todescribe relationships between quantities that vary in a functional relationship. In the remainingparts of this investigation, well explore function notation as we look at other growth patterns and

situations.

2. Fiona has a younger brother, Tyler, who attends a pre-kindergarten class for 4-year olds. Oneof the math activities during the first month of school was measuring the heights of thechildren. The class made a large poster to record the information in a bar graph that had a barfor the height of each child. The children used heights rounded to the nearest whole number ofinches; however, the teacher also used an Excel spreadsheet to record their heights to thenearest half inch, as shown below.

Table A: Class Height Information

August 2007

StudentNumber Last name First name

Height ininches

1 Barnes Joshua 42.0

2 Coleman David 40.5

3 Coleman Diane 40.0

4 Drew Keisha 37.5

5 Evans Tyler 39.5

6 Hagan Emily 38.0

7 Nguyen Violet 37.0

8 Rader Joshua 38.5

9 Ruiz Alina 38.5

10 Vogel Zach 39.5

After making the Excel table, the teacher decided to also make an Excel version of the bargraph. While she was working on the bar graph, she had the idea of also graphing theinformation in the rectangular coordinate system, using the Student Number as the x-value andthe height to the nearest half inch as the y-value. Here is her graph.


8/61





Unit 1: Page8 of61

The relationship that uses Student Number as input and the height of the student with thatstudent number as output describes a function because, for each student number, there isexactly one output, the height of the student with that student number.

a. The graph was drawn with an Excel option named scatter plot. This option allows graphsof relationships whether or not the graphs represent functions. Sketch a scatter plot usingstudent last names as inputs and heights of students as outputs. Explain why thisrelationship is not a function.

b. Sketch a scatter plot using student first names as inputs and heights of students as outputs.Is this relationship a function? Explain your answer.

c. In Graph A, the pre-kindergarten teacher chose an Excel format that did not connect thedots. Explain why the dots should not be connected.

d. Using the information in the Excel spreadsheet about the relationship between student

number and height of the corresponding student, fill in each of the following blanks.

The height of student 2 is equal to _____inches.The height of student 6 is equal to _____ inches.The height of student _____ is equal to 37 inches.For what student numbers is the height equal to 38.5 inches? _____


9/61





Unit 1: Page9 of61

e. We are now ready to discuss function notation. First we need to give our function amathematical name. Since the outputs of the functions are heights, we name the function h.In Mathematics I and following courses, well use one letter names for functions as a way

to refer to the whole relationship between inputs and outputs. So, for this example, wemean that h consists of all the input-output pairs of student number and correspondingheight in inches; thus, we can say that Graph A above is the graph of the function hbecause the graph shows all of these input-output pairs.

Using the function name h, we write h(1) = 42 to indicate that the height of student 1 isequal to 42 inches. We read h(1) as hof 1. This wording is similar tothe phrase heightof student 1. Without referring to the specific meaning of function h, we say that h(1)means the output value of function h when the input value is 1. Since the output value isthe number 42, we write h(1) = 42.

More examples: h(3) = 39 (read hof 3 equals 39) means that student 3 is 39 in. tall,and h(4) = 37.5 (read hof 4 equals 37.5) means that student 4 is 37.5 in. tall.

The following fill-in-the-blank questions repeat the questions from part d) in functionnotation. Fill in these blanks too.

h(2) = _____ h(6) = _____ h(____) = 37

For what values ofx does h(x) = 38.5 ? _____

3. We now return to the function in #1 above and name this functionJ(for Julias height).Consider the notationJ(2). We note that function notation gives us another way to write aboutideas that you began learning in middle school, as shown in the table below.

Statement Type

At age 2, Julia was 35 inches tall. Natural languageWhenx is 2,y is 35. Statement about variablesWhen the input is 2, the output is 35. Input-output statementJ(2) = 35. Function notation

The notationJ(x) is typically read Jofx, but thinking Jatx is also useful sinceJ(2) can beinterpreted as height at age 2, for example.

Note: Function notation looks like a multiplication calculation, but the meaning is verydifferent. To avoid misinterpretation, be sure you know which letters represent functions. Forexample, ifg represents a function, then g(4) isnot multiplication of g and 4 but is rather thevalue of gat 4, that is, the output value of the function g when the input is value is 4.


10/61





Unit 1: Page10 of61

a. What isJ(11)? What does this mean?

b.

Whenx is 3, what isy? Express this fact using function notation.

c. Find anx so thatJ(x) = 53. Explain your method. What does your answer mean?

d. From your graph or your table, estimateJ(6.5). Explain your method. What does youranswer mean?

e. Estimate a value forx so thatJ(x) is approximately 60. Explain your method. What does

your answer mean?

f. Describe what happens toJ(x) asx increases from 0 to 16.

g. What can you say aboutJ(x) forx greater than 16?

h. Describe the similarities and differences you see between these questions and the questionsin #1.

4. Fiona attends Peachtree Plains High School. When the school opened five years ago, a fewteachers and students put on Fall Fest, featuring contests, games, prizes, and performances bystudent bands. To raise money for the event, they sold Fall Fest T-shirts. The event was verywell received, and so Fall Fest has become a tradition. This year Fiona is one of the studentshelping with Fall Fest and is in charge of T-shirt sales. She gathered information about thegrowth of T-shirt sales for the Fall Fests so far and created the graph below that shows thefunction S.


11/61





Unit 1: Page11 of61

70

60

50

40

30

20

10

2 4

S(t),

# ofshirts

t, in years

a. What are the independent and dependent variables shown in the graph?

b. For which years does the graph provide data?

c. Does it make sense to connect the dots in the graph? Explain.

d. What were the T-shirt sales in the first year? Use function notation to express your result.

e. Find S(3), if possible, and explain what it means or would mean.

f. Find S(6), if possible, and explain what it means or would mean.

g. Find S(2.4), if possible, and explain what it means or would mean.

h. If possible, find a tsuch that S(t) = 65. Explain.

i. If possible, find a tsuch that S(t) = 62. Explain.

j. Describe what happens to S(t) as tincreases, beginning at t= 1.

k. What can you say about S(t) for values oftgreater than 6?


12/61





Unit 1: Page12 of61

Note: As you have seen above, functions can be described by tables and by graphs. In high schoolmathematics, functions are often given by formulas. In the remaining items for this task, wedevelop, or are given, a formula for each function under consideration, but it is important to

remember that not all functions can be described by formulas.

5. Fionas T-shirt committee decided on a long sleeve T-shirt in royal blue, one of the schoolcolors, with a FallFest logo designed by the art teacher. The committee needs to decide howmany T-shirts to order. Fiona was given the job of collecting price information so she checkedwith several suppliers, both local companies and some on the Web. She found the best price atPeachtree Plains Promotions, a local company owned by parents of a Peachtree Plains HighSchool senior.

The salesperson for Peachtree Plains Promotions told Fiona that there would be a $50 fee forsetting up the imprint design and different charges per shirt depending on the total number of

shirts ordered. For an order of 50 to 250 T-shirts, the cost is $9 per shirt. Based on sales fromthe previous five years, Fiona is sure that they will order at least 50 T-shirts and will not ordermore than 250. Ifx is the number of T-shirts to be ordered for this years FallFest, andy is thetotal dollar cost of these shirts, theny is a function ofx. Lets name this function C, for costfunction. Fiona started the table below.

x 50 100 150 200 250

9x 450 900 1350y = C(x) 500 950

a. Fill in the missing values in the table above.

b.

Make a graph to show how the cost depends upon the number of T-shirts ordered. You canstart by plotting the points corresponding to values in the table. What points are these?Should you connect these points? Explain. Should you extend the graph beyond the firstor last point? Explain.

c. Write a formula showing how the cost depends upon the number of T-shirts ordered. Forwhat numbers of T-shirts does your formula apply? Explain.

d.

What does C(70) mean? What is the value of C(70)? Did you use the table, the graph, orthe formula?

e. If the T-shirt committee decides to order only the 67 T-shirts that are pre-paid, how muchwill it cost? Show how you know. Express the result using function notation.

f. If the T-shirt committee decides to order the 67 T-shirts that are pre-paid plus 15 more,how much will it cost? Show how you know. Express the result using function notation.


13/61





Unit 1: Page13 of61

6. Fiona is taking physics. Her sister, Hannah, is taking physical science. Fiona decided touse functions to help Hannah understand one basic idea related to gravity and fallingobjects. Fiona explained that, if a ball is dropped from a high place, such as the Tower of

Pisa in Italy, then there is a formula for calculating the distance the ball has fallen. Ify,measured in meters, is the distance the ball has fallen and x, measured in seconds, is thetime since the ball was dropped, theny is a function ofx, and the relationship can beapproximated by the formulay = d(x) = 5x

2. Here we name the function dbecause the

outputs are distances.

x 0 1 2 3 4 5 6 x

2 0 1 4 9

y = d(x) = 5x2

0 5 20

a. Fill in the missing values in the table above.

b. Suppose the ball is dropped from a building at least 100 meters high. Measuring from thetop of the building, draw a picture indicating the position of the ball at times indicated inyour table of values.

c. Draw a graph ofx versusy for this situation. Should you connect the dots? Explain.

d. What is the relationship between the picture (part b) and the graph (part c)?

e. You know from experience that the speed of the ball increases as it falls. How can yousee the increasing speed in your table? How can you see the increasing speed in yourpicture?

f. What is d(4)? What does this mean?

g. Estimatex such that d(x) = 50. Explain your method. What does it mean?

h. In this context,y is proportional tox2. Explain what that means. How can you see this in

the table?


14/61





Unit 1: Page14 of61

7. Fiona is paid $7 per hour in her part-time job at the local Dairy Stop. Let tbe the amount timethat she works, in hours, during the week, and let P(t) be her gross pay (before taxes), indollars, for the week.

a.

Make a table showing how her gross pay depends upon the amount of time she worksduring the week.

b. Make a graph illustrating how her gross pay depends upon the amount of time that sheworks. Should you connect the dots? Explain.

c. Write a formula showing how her gross pay depends upon the amount of time that sheworks.

d. What is P(9)? What does it mean? Explain how you can use the graph, the table, and the

formula to compute P(9).

e. If Fiona works 11 hours and 15 minutes, what will her gross pay be? Show how youknow. Express the result using function notation.

f. If Fiona works 4 hours and 50 minutes, what will her gross pay be? Show how you know.Express the result using function notation.

g. One week Fionas gross pay was $42. How many hours did she work? Show how you

know.

h. Another week Fionas gross pay was $57.19. How many hours did she work? Show howyou know.


15/61





Unit 1: Page15 of61

Fences and Functions Learning Task

1. Claire decided to plant a rectangular garden in her back yard using 30 pieces of fencing that

were given to her by a friend. Each piece of fencing was a vinyl panel 1 yard wide and 6 feethigh. Claire wanted to determine the possible dimensions of her garden, assuming that shewould use all of the fencing and did not cut any of the panels. She began by placing ten panels(10 yards) parallel to the back side of her house and then calculated that the other dimension ofher garden then would be 5 yards, as shown in the diagram below.

Claire looked at the 10 fencing panels laying on the ground and decided that she wanted toconsider other possibilities for the dimensions of the garden. In order to organize her thoughts,she letx be the garden dimension parallel to the back of her house, measured in yards, and letybe the other dimension, perpendicular to the back of the house, measured in yards. Sherecorded the first possibility for the dimensions of the garden as follows: Whenx = 10,y = 5.

a. Explain whyy must be 5 whenx is 10.

b. Make a table showing the possibilities forx andy.

c. Find the perimeter of each of the possible gardens you listed in part b. What do younotice? Explain why this happens.

d. Did you considerx = 15 in part b? Ifx = 15, what must y be? What would Claire do withthe fencing if she chosex = 15?

e. Canx be 16? What is the maximum possible value forx? Explain.


16/61





Unit 1: Page16 of61

f. Write a formula relating they-dimension of the garden to thex-dimension.

g. Make a graph of the possible dimensions of Claires garden.

What would it mean to connect the dots on your graph? Does connecting the dots makesense for this context? Explain.

h.

As thex-dimension of the garden increases by 1 yard, what happens to they-dimension?Does it matter whatx-value you start with? How do you see this in the graph? In thetable? In your formula? What makes the dimensions change together in this way?

2. After listing the possible rectangular dimensions of the garden, Claire realized that she neededto pay attention to the area of the garden, because area determines how many plants can begrown.

a. Does the area of the garden change as thex-dimension changes? Make a prediction, andexplain your thinking.

b. Use grid paper to make accurate sketches for at least three possible gardens. How is thearea of each garden represented on the grid paper?

c. Make a table listing all the possiblex-dimensions for the garden and the correspondingareas. (To facilitate your calculations, you might want to include they-dimensions in yourtable or add an area column to your previous table.)

d. Make a graph showing the relationship between thex-dimension and the area of the garden.Should you connect the dots? Explain.


17/61





Unit 1: Page17 of61

3. Because the area of Claires garden depends upon thex-dimension, we can say that the area isa function of thex-dimension. Lets use G for the name of the function that uses eachx-dimension an input value and gives the resulting garden area as the corresponding output

value.a. Use function notation to write the formula for the garden area function. What does G(11)mean? What is the value ofG(11)? What line of your table, from #2, part c, and whatpoint on your graph, from #2, part d, illustrate this same information?

b. The set of all possible input values for a function is called thedomainof the function.What is the domain of the garden area function G? How is the question about domainrelated to the question about connecting the dots on the graph you drew for #2, part d?

c. The set of all possible output values is called therange of the function. What is the rangeof the garden area function G? How can you see the range in your table? In your graph?

d. As thex-dimension of the garden increases by 1 yard, what happens to the garden area?Does it matter whatx-dimension you start with? How do you see this in the graph? In thetable? Explain what you notice.

e. What is the maximum value for the garden area, and what are the dimensions when thegarden has this area? How do you see this in your table? In your graph?

f. What is the minimum value for the garden area, and what are the dimensions when thegarden has this area? How do you see this in your table? In your graph?

g. In deciding how to lay out her garden, Claire made a table and graph similar to those youhave made in this investigation. Her neighbor Javier noticed that her graph had symmetry.Your graph should also have symmetry. Describe this symmetry by indicating the line ofsymmetry. What about the context of the garden situation causes this symmetry?


18/61





Unit 1: Page18 of61

h. After making her table and graph, Claire made a decision, put up the fence, and planted hergarden. If it had been your garden, what dimensions would you have used and why?

4. Later that summer, Claires sister-in-law Kenya mentioned that she wanted to use 30 yards ofchain-link fence to build a new pen for her two pet pot bellied pigs. Claire experienced dj vuand shared how she had analyzed how to fence her garden. As Claire explained her analysis,Kenya realized that her fencing problem was very similar to Claires.

a. Does the formula that relates they-dimension of Claires garden to thex-dimension of hergarden apply to the pen for Kenyas pet pigs? Why orwhy not?

b.

Does the formula that relates the area of Claires garden to itsx-dimension apply to the penfor Kenyas pet pigs? Why or why not?

c. Write a formula showing how to compute the area of the pen for Kenyas pet pigs given thex-dimension for the pen.

d. Does thex-dimension of the pen for Kenyas pet pigs have to be a whole number? Explain.

e. Let P be the function which uses the x-dimension of the pen for Kenyas pet pigs as inputand gives the area of the pen as output. Write a formula for P(x).

f. Make a table of input and output values for the function P. Include some values of thex-dimension for the pen that that could not be used as thex-dimension for Claires garden.

g. Make a graph of the function P.


19/61





Unit 1: Page19 of61

h. Does your graph show any points with x-value less than 1? Could Kenya have made a penwith yard as thex-dimension? If so, what would the other dimension be? How about apen with anx-dimension 0.1 of a yard? How big is a pot bellied pig? Would a pot bellied

pig fit into either of these pens?

i. Is it mathematically possible to have a rectangle withx-dimension equal to ? How abouta rectangle withx-dimension 0.1?

j. Of course, Kenya would not build a pen for her pigs that did not give enough room for thepigs to turn around or pass by each other. However, in analyzing the function P to decide

how to build the pen, Kenya found it useful to consider all the input values that could bethex-dimension of a rectangle. She knew that it didnt make sense to consider a negativenumber as thex-dimension for the pen for her pigs, but she asked herself if she couldinterpret anx-dimension of 0 in any meaningful way. She thought about the formularelating they-dimension to thex-dimension and decided to includex = 0. What layout offencing would correspond to the valuex = 0? What area would be included inside thefence? Why could the shape created by this fencing layout be called a degeneraterectangle?

k.

The valuex

= 0 is called alimiting case

. Is there any other limiting case to consider inthinking about values for thex-dimension of the pen for Kenyas pigs? Explain.

l. Return to your graph of the function P. Adjust your graph to include all the values thatcould mathematically be thex-dimension of the rectangular pen even though some of theseare not reasonable for fencing an area for pot bellied pigs.

(Note: You can plot points corresponding to any limiting cases for the function using asmall circle. To include the limiting case,fill in the circle to make a solid dot. To notinclude the limiting case, but just use it to show that the graph does not continue beyondthat point, leave the circle openO.)

5. Use your table, graph, and formula for the function P to answer questions about the pen forKenyas pet pigs.


20/61





Unit 1: Page20 of61

a. Estimate the area of a pen of with anx-dimension of 10 feet (not yards). Explain yourreasoning.

b. Estimate thex-dimension of a rectangle with an area of 30 square yards. Explain yourreasoning.

c. What is the domain of the function P? How do you see the domain in the graph?

d. What point on the graph corresponds to the pen with maximum area?

e. What is the maximum area that the pen for Kenyas pigs could have? Explain. What doyou notice about the shape of the pen?

f. What is the range of the function P? How can you see the range in the graph?

g. How should you move your finger along thex-axis (right-to-left or left-to-right) so that thex-value increases as your move your finger? If you move your finger along the graph ofthe function P in this same direction, do thex-values of the points also increase? As thex-dimension of the pen for Kenyas pigs increases, sometimes the area increases andsometimes the area decreases. Using your graph, determine thex-values such that the areaincrease asx increases. For whatx-values does the area decrease asx increases?

When using tables and formulas, we often look at a function by examining only one or two pointsat a time, but in high school mathematics, it is important to begin to think about the wholefunction, that is, all of the input-output pairs. Weve started working on this idea already byusing a single letter such asf, G, or P to refer to the whole collection of input-output pairs. Wellgo further as we proceed with this investigation.


21/61





Unit 1: Page21 of61

We say that two functions are equal (as whole functions) if they have exactly the same input-output pairs. In other words, two functions are equal if they have the same domainand the outputvalues are the same for each input value in the domain . From a graphical perspective, two

functions are equal if their graphs have exactly the same points. Note that the graph of a functionconsists of all the points which correspond to input-output pairs, but when we draw a graph weoften can show only some of the points and indicate the rest. For example, if the graph of thefunction is a line, we show part of the line and use arrowheads to indicate that the line continueswithout end.

6. The possibilities for the pen for Kenyas pet pigs andfor Claires garden are very similar insome respects but different in others. These two situations involve differentfunctions, eventhough theformulas are the same.

a. If Kenya makes the pen with maximum area, how much more area will the pen for her pet

pigs have than Claires garden of maximum area? How much area is that in square feet?

b. What could Claire have done to build her garden with the same area as the maximum areafor Kenyas pen? Do you think this would have been worthwhile?

c. Consider the situations that led to the functions G and P and review your tables, graphs,

formulas related to the two functions. Describe the similarities and differences betweenKenyas pig pen problem and Claires garden problem. Your response should include adiscussion ofdomain and range for the two functions.


22/61





Unit 1: Page22 of61

From Wonderland to Functionland Learning Task

Consider the following passage from Lewis CarrollsAlices Adventures in Wonderland, ChapterVII, A Mad Tea Party.

"Then you should say what you mean." the March Hare went on.

"I do," Alice hastily replied; "at least -- at least I mean what I say -- that's the same thing,you know."

"Not the same thing a bit!" said the Hatter, "Why, you might just as well say that 'I seewhat I eat' is the same thing as 'I eat what I see'!"

"You might just as well say," added the March Hare, "that 'I like what I get' is the samething as 'I get what I like'!"

"You might just as well say," added the Dormouse, who seemed to be talking in his sleep,"that 'I breathe when I sleep' is the same thing as 'I sleep when I breathe'!"

"It is the same thing with you," said the Hatter, and here the conversation dropped, and theparty sat silent for a minute.

Lewis Carroll, the author ofAlice in Wonderlandand Through the Looking Glass, was amathematics teacher who had fun playing around with logic. In this activity, youll investigate

some basic ideas from logic and perhaps have some fun too.

We need to start with some basic definitions.

Astatement is a sentence that is either true or false, but not both.

Aconditional statementis a statement that can be expressed in if then form.

A few examples should help clarify these definitions. The following sentences are statements.

Atlanta is the capital of Georgia. (This sentence is true.)

Jimmy Carter was the thirty-ninth president of the United States and was born in Plains,

Georgia. (This sentence is true.)The Atlanta Falcons are a professional basketball team. (This sentence is false.)

George Washington had eggs for breakfast on his fifteenth birthday. (Although it is unlikelythat we can find any source that allows us to determine whether this sentence is true or false, it

still must either be true or false, and not both, so it is a statement.)

Here are some sentences that are not statements.


23/61





Unit 1: Page23 of61

Whats your favorite music video? (This sentence is a question.)

Turn up the volume so I can hear this song. (This sentence is a command.)

This sentence is false. (This sentence is a very peculiar object called a self-referential

sentence. It creates a logical puzzle that bothered logicians in the early twentieth century.If the sentence is true, then it is also false. If the sentence is false, then it is not false and,

hence, also true. Logicians finally resolved this puzzling issue by excluding such sentences

from the definition of statement and requiring that statements must be either true orfalse, but not both.)

The last example discussed a sentence that puzzled logicians in the last century. The passage fromAlice in Wonderlandcontains several sentences that may have puzzled you the first time you readthem. The next part of this activity will allow you to analyze the passage while learning moreabout conditional statements.

Near the beginning of the passage, the Hatter responds to Alice that she might as well say that Isee what I eat means the same thing as I eat what I see. Lets express each of the Hattersexample sentences in if then form.

I see what I eat has the same meaning as the conditional statement If I eat a thing, then I see it.On the other hand, I eat what I see has the same meaning as the conditional statement If I see athing, then I eat it.

1. Express each of the following statements from the Mad Tea Party in if then form.

a. I like what I get. ______________________________________________

b. I breathe when I sleep. ______________________________________________

2. We use specific vocabulary to refer to the parts of a conditional statement written in if then form. Thehypothesis of a conditional statement is the statement that follows the wordif. So, for the conditional statement If I eat a thing, then I see it, the hypothesis is thestatement I eat a thing. Note that the hypothesis doesnotinclude the word if because thehypothesis is the statement that occurs after the if.

Give the hypothesis for each of the conditionals in item 1, parts a and b.

3. Theconclusion of a conditional statement is the statement that follows the word then. So,for the conditional statement If I eat a thing, then I see it, the conclusion is the statement Isee it. Note that the conclusion does notinclude the word then because the conclusion isthe statement that occurs after the word then.


24/61





Unit 1: Page24 of61

Give the conclusion for each of the conditionals in item 1, parts a and b.

Now, lets get back to the discussion at the Mad Tea Party. When we expressed the Hatters

example conditional statements in if then form, we used the pronoun it in the conclusion ofeach statement rather than repeat the word thing. Now, we want to compare the hypotheses(note that the word hypotheses is the plural of the word hypothesis) and conclusions of theHatters conditionals. To help us see the key relationship between his two conditional statements,we replace the pronoun it with the noun thing. This replacementdoesnt change the meaning.As far as English prose is concerned, we have a repetitious sentence. However, this repetitionhelps us analyze the relationship between hypotheses and conclusions.

4. a. List the hypothesis and conclusion for the revised version of each of the Hatters conditionalstatements given below.

Hypothesis Conclusion

If I eat a thing, then I see the thing. ________________ ________________

If I see a thing, then I eat the thing. ________________ ________________

b. Explain how the Hatters two conditional statements are related.

5. There is a term for the new statement obtained by exchanging the hypothesis and conclusion ina conditional statement. This new statement is called theconverse of the first.

a. Write the converse of each of the conditional statements in item1, parts a and b, using if then form.

b. What happens when you form the converse of each of the conditional statements given asanswers for this item part a?

6. The March Hare, Hatter, and Dormouse did not use if then form when they stated theirconditionals. Write the converse for each conditional statement below without using if then form.

Conditional: I breathe when I sleep. Converse: ________________________________

Conditional: I like what I get. Converse: ________________________________

Conditional: I see what I eat. Converse: ________________________________

Conditional: I say what I mean. Converse: ________________________________


25/61





Unit 1: Page25 of61

7. a. What relationship between breathing and sleeping is expressed by the conditional statementI breathe when I sleep? If you make this statement, is it true or false?

c. What is the relationship between breathing and sleeping expressed by the conditionalstatement I sleep when I breathe? If you make this statement, is it true or false?

8. The conversation in passage fromAlice in Wonderlandends with the Hatters response to theDormouse "It is the same thing with you." The Hatter was making a joke. Do you get the

joke? If you arent sure, you may want to learn more about Lewis Carrolls characterization ofthe Dormouse in Chapter VII ofAlice in Wonderland.

We want to come to some general conclusions about the logical relationship between aconditional statement and its converse. The next steps are to learn more of the vocabulary fordiscussing such statements and to see more examples.

In English class, you learn about compound sentences. As far as English is concerned, compoundsentences consist two or more independent clauses joined by using a coordinating conjunctionsuch as and, or, but, and so forth, or by using a semicolon.

In logic, the term compound is used in a more general sense. Acompound statement, orcompound proposition, is a new statement formed by putting two or more statements together toform a new statement. There are several specific ways to combine statements to create acompound proposition. Compound statements formed using and and or are important in thestudy of probability. In this task, we are focusing on compound propositions created using the if then form. In order to talk about this type of compound proposition without regard to theparticular statements used for the hypothesis and conclusion, we can use variables to representstatements as a whole. This use of variables is demonstrated in the formal definition that follows.


26/61





Unit 1: Page26 of61

Definition: Ifp and q are statements, then the statement ifp, then q is theconditionalstatement, or implication, with hypothesisp and conclusion q.

We call the variables used above,statement, orpropositional, variables. We seek a generalconclusion about the logical relationship between a conditional statement and its converse; we arelooking for a relationship that is true no matter what particular statements we substitute for thestatement variablesp and q. Thats why we need to see more examples.

Our earlier discussions of function notation, domain of a function, and range of a function haveincluded conditional statements about inputs and outputs of a function. For the next part of thisactivity, we consider conditional statements about a particular function, theabsolute valuefunctionf, defined as follows:

f is the function with domain all real numbers such thatf(x) = |x |.

(Note. To give a complete definition of the absolute value function, we must specify the domainand a formula for obtaining the unique output for each input. It is not necessary to specify therange because the domain and the formula determine the set of outputs.)

9. Well explore the graph of the absolute value functionfand then consider some related

conditional statements.a. Complete table of values given below.

x 01

2

1

2 1 1 3.8 3.8 5

f(x) = |x | 01

2

1

2 5

15

2

15

2 10

d. Most graphing calculators have a standard graphing window which shows the portion of

the graph of a function corresponding tox-values from10 to 10 andy-values from10 to10. On grid paper, set up such a standard viewing window and then use the table of valuesabove to draw the part of the graph of the functionffor this viewing window. Does yourgraph show all of the input/output pairs listed in your table of values? Does your graphshow the input/output pairs for thosex-values from10 to 10 that were not listed in thetable? Explain.


27/61





Unit 1: Page27 of61

c. Think of your graph as a picture. Do you see a familiar shape? Describe the graph tosomeone who cannot see it. If you were to extend your graph to include pointscorresponding to additional values ofx, say to includex-values from1000 to 1000, would

the shape change? What can you do to your sketch to indicate information about thepoints outside of your viewing window?

e. For whatx-values shown on your graph offdoes they-value increase asx increases, that is,for whatx-values is it true that, as you move your finger along your graph so that thex-values increase, then they-values also increase? (Be sure to give thex-values where thishappens.) For whatx-values does they-value decrease asx increases, that is, for whatx-values is it true that, as you move your finger along your graph so that thex-valuesincrease, then they-values decrease? (Be sure to give thex-values where this happens.)Considering your answers to the questions in part c, for the whole functionf, determine (i)

thosex-values such that they-value increases asx increases and (ii) thosex-values suchthat they-value decreases asx increases.

10. Evaluate each of the following expressions written in function notation. Be sure to simplify sothat there are no absolute value signs in your answers. Use your graph to verify that each ofyour statements is true.

a. f(0) = ____ b. f(5) =____ c. f( ) = ____ d. f( 3 ) = ____

11. The statements in 10, parts ad, are written using the equals sign. The same ideas can beexpressed with conditional statements. Fill in the blanks to form such equivalent statements.

a. Ifx = 0, thenf(x) = ____. b. Ifx =5, thenf(x) = ____ .

c. If the input for the functionfis , then the output for the functionfis ____ .

d. If the input for the functionfis 3 , then the output for the functionfis ____.

In the remainder of this task, we will often consider whether a conditional statement is true or

false. To say that a conditional is true means that, whenever the hypothesis is true, then theconclusion is also true; and to say that a conditional is false means that the hypothesis is, or canbe, true while the conclusion is false.


28/61





Unit 1: Page28 of61

12. a. Write the converse of each of the true conditional statements from item #11. For eachconverse, use the graph off to determine whether the statement is true or false. Organizeyour work in a table such as the one shown below. For the statements in the table that you

classify as false, specify a value ofx that makes the hypothesis true and the conclusionfalse.

Conditional statement Truthvalue

Converse statement Truth value

Ifx = 0, thenf(x) = ___. True

True

True

True

b. As indicated by the header line in the table above, whether a statement is true or false iscalled thetruth value of the statement. Our goal for this item is to decide whether there isa general relationship between the truth value of a conditional statement and the truth valueof its converse. Any particular conditional statement can be true or false, so we need toconsider examples for both cases. Add lines to your table from part a for the converses ofthe following false conditional statements. For these statements, and for any converse thatyou classify as false, give a value ofx that makes the hypothesis true and the conclusionfalse.

(i) Iff(x) = 7, thenx = 5. (ii) Iff(x) = 2, thenx = 2.

Conditional statement Truth value Converse statement Truth value


29/61





Unit 1: Page29 of61

c. Complete the following sentence to make a true statement. Explain your reasoning. Isyour answer choice consistent with all of the examples of converse in the table above?

Multiple choice: The converse of a true conditional statement is ____.A) always also true B) always false

C) sometimes true and sometimes false because whether the converse is true or falsedoes not depend on whether the original statement is true or false.

d. Complete the following sentence to make a true statement. Explain your reasoning. Isyour answer choice consistent with all of the examples of converse in the table above?

Multiple choice: The converse of a false conditional statement is ____.

A) always also false B) always trueC) sometimes true and sometimes false because whether the converse is true or false

does not depend on whether the original statement is true or false.

Whenever we talk about statements in general, without having a particular example in mind, itis useful to talk about thepropositional formof the statement. For the propositional form ifp, then q, the converse propositional form is ifq, thenp.

If two propositional forms result in statements with the same truth value for all possible casesof substituting statements for the propositional variables, we say that the forms are logically

equivalent. If there exist statements that can be substituted into the propositional forms so thatthe resulting statements have different truth values, we say that the propositional forms arenotlogically equivalent

e. Consider your answers to parts a and b, and decide how to complete the followingstatement to make it true. Justify your choice.

The conversepropositional form ifq, then p is/is not (choose one) logicallyequivalent to the conditional statement ifp, then q.

f. Multiple choice: If you learn a new mathematical fact in the form ifp, then q, what canyou immediately conclude, without any additional information, about the truth value of theconverse?

A) no conclusion because the converse is not logically equivalent

B) conclude that the converse is true C) conclude that the converse is false


30/61





Unit 1: Page30 of61

g. Look back at the opening of the passage fromAlice in Wonderland, when Alice hastilyreplied "I do, at least -- at least I mean what I say -- that's the same thing, you know."What statements did Alice think were logically equivalent? What was the Hatter saying

about the equivalence of these statements when he replied to Alice by saying "Not the samething a bit!"?

There are two other important propositional forms related to any given conditional statement. Weintroduce these by exploring other inhabitants of the land of functions.

13.Let g be the function with domain all real numbers such that g(x) = |x | + 3 .

a. Complete table of values given below.

x 01

2

1

2 1 1 3.8 3.8 5

g(x) = |x | + 3 37

2

7

2 8

21

2

21

2 13

b. On grid paper, draw the portion of the graph ofg for allx-values such that 10 10x . To

show all of the points corresponding to input/output pairs shown in the table, how much ofthey-axis should your viewing window include? Is the part of the graph that you havedrawn representative of the whole graph? Explain.

c. For whatx-values shown on your graph ofg does they-value increase asx increases? Forwhatx-values does they-value decrease asx increases? For the whole function g,determine (i) thosex-values such that they-value increases asx increases and (ii) thosex-values such that they-value decreases asx increases.

d. What is the relationship between the graphs off and g? What in the formulas forf(x) andg(x) tells you that the graphs are related in this way?


31/61





Unit 1: Page31 of61

14.It is clear from the graphs offand g that, for each input value, the two functions have differentoutput values. For just one example, we see thatf(4) = 4 but g(4) = 7. If we want toemphasize that g is not the absolute value function and that g(4) is different from 4, we could

write g(4) 4, which is read g of 4 is not equal to 4. We now examine some relatedconditional statements.

a. Complete the following conditional statement to indicate that g(4) 4.

If the input of the function g is 4, then the output of the function g is not ____.

b. Consider another true statement about the function g; in this case the statement is g(3) 5. Use your graph to evaluate g(3) and verify that g(3) 5 is a true statement.

Letprepresent the statement The input of the function g is3.Let qrepresent the statement The output of the function g is not 5.What statement is represented by ifp, then q? Does this statement express the idea thatg(3) 5?

In logic, we form thenegation of a statementpby forming the statement It is not true thatp.For convenience, we use notp to refer to the negation of the statementp. For a specificchoice of statement, when we translate notp into English, we can usually state the negationin a more direct way. For example:

when p represents the statement The input of the function g is3,

then notp representsThe input of the function g is not3, and

when, as above, q represents the statement The output of the function g is not 5,

then not qrepresents The output of the function g is not not 5, or more simplyThe output of the function g is 5.

c. Ifp and q are represent the statements indicated in part b:

(i) What statement is represented by If notp, then not q?

(ii) Is this inverse statement true? Explain your reasoning.

(iii) Does this inverse statement tell you that g(3) 5?


32/61





Unit 1: Page32 of61

A statement of the form If notp, then not q is called the inverse of the conditional statementifp, then q. Note that the inverse is formed by negating the hypothesis and conclusion of aconditional statement.

d. The table below includes statements about the functionsfand g. Fill in the blanks in thetable. Be sure that your entries for the truth value columns agree with the graphs forfandg. For the statements in the table that are false, give a value ofx that makes the hypothesistrue and the conclusion false.

Conditional statement Truthvalue

Inverse statement Truth value

Ifx = 4, thenf(x) 9. True Ifx 4, thenf(x) = 9.

Ifg(x) 3, then x 0.

Ifx = 0, then g(x) 3.

Ifg(x) 6, thenx 3. True

e. Consider the results in the table above, and then decide how to complete the followingstatement to make it true. Justify your choice.

The inversepropositional form ifnotp, then not q is/ is not (choose one) logically

equivalent to the conditional statement ifp, then q.

f. Multiple choice: If you learn a new mathematical result in the form ifp, then q, what canyou immediately conclude, without any additional information, about the truth value of theinverse?

A) no conclusion because the inverse is not logically equivalentB) conclude that the inverse is true C) conclude that the inverse is false

15.Let h be the function with domain all real numbers such that h(x) = 2|x | .

a. On grid paper, draw the portion of the graph ofh for allx-values such that 10 10x .

b. For x =7,2, 0, 5, 8, comparef(x) and h(x). What is the relationship between the values?Does this relationship hold for every real number? How do the graphs offand h compare?Explain why the graphs have this relationship.


33/61





Unit 1: Page33 of61

c. When any positive real number is used as the input for the absolute value functionf, thenthe output is the same as the input.For example,f(1) = 1,f(4) = 4,f(10) = 10, and so forth. However, the function h does not

have this property. For example, h(10) 10. Express this inequality in two different waysby completing the following conditional statements.

(i) If the input of the function h is 10, then the output of the function h is not ________.

(ii) If the output of the function h is 10, then the input of the function h________.

d. For the conditional statement in (i), part c, above, let p denote the hypothesis of thestatement, and let q denote the conclusion. Then, express the propositional form of thestatement (ii), part c, using notp and not q.

A statement of the form If not q, then notp is called thecontrapositive of the conditionalstatement ifp, then q. Note that the contrapositive is formed by both negating andexchanging the hypothesis and conclusion.

e. The table below includes statements about the function h. Fill in the blanks in the table.Be sure that your answers for the truth value columns are consistent with the graph ofh.For the statements in the table classified as false, give a value ofx that makes thehypothesis true and the conclusion false.

Conditional statement Truth value Contrapositive statement Truth value

Ifx = 4, then h(x) = 3. Ifh(x) 3, thenx 4.

Ifx = 6, then h(x) = 12.

Ifh(x) = 0, thenx = 0.

Ifx < 3, then h(x) < 6.False,

let x =4

f. On each line of the table, how do the truth value of the conditional statement and itscontrapositive compare?


34/61





Unit 1: Page34 of61

16.a. Suppose that you are given that ifp, then q is a true statement for some particular choiceofp and q. For example, suppose that there is a function kwhose domain and range are allreal numbers and it is true that, if the input to the function kis 17, then the output of

function kis86. What is the hypothesis of the contrapositive statement? What is theconclusion of the contrapositive statement? Given that k(17) =86, is the contrapositivetrue? Explain.

b. Suppose that you are given that the contrapositive statement if not q, then notp is a truestatement for some particular choice ofp and q. For example, suppose that there is afunction kwhose domain and range are all real numbers and it is true that, if the output ofthe function kis not 101, then the input to the function kis not 34.If we regard If the output of the function kis not 101, then the input to the function kis

not 34 as the contrapositive statement, what was the original conditional statement? If weare given that the contrapositive statement is true, must the original conditional be true.Explain.

c. Based on your reasoning in parts a and b, if a conditional statement is true, could thecontrapositive be false? If a conditional statement is false, could the contrapositivestatement be true?

d. Consider your answer to part c, and decide how to complete the following statement tomake it true. Justify your choice.

The propositional form ifp, then q is/ is not (choose one) logically equivalent to itscontrapositive if not q, then notp.

e. Multiple choice: If you learn a new mathematical result in the form ifp, then q, what canyou immediately conclude, without any additional information, about the truth value of thecontrapositive?

A) no conclusion because the contrapositive is not logically equivalent

B) conclude that the contrapositive is true C) conclude that the contrapositive is false


35/61





Unit 1: Page35 of61

17.Summarizing the information about forming related conditional statements, we see that theconditional ifp, then q, has three related conditional statements:converse: ifq, thenp (swaps the hypothesis/conclusion)

inverse: if notp, then not q (negates the hypothesis/conclusion)

contrapositive: if not q, then notp (negates hypothesis/conclusion, then swaps)

Which, if any, of these is logically equivalent to the original conditional statement and alwayshas the same truth value as the original? ________________________

The absolute value functionf and the functions g and h that you have worked with in thisinvestigation are not linear. However, in your study of functions prior to Mathematics I, youhave worked with many linear functions. We conclude this investigation with discussion about

converse, inverse, and contrapositive using a linear function.

18.Consider the linear function Fwhich converts a temperature ofc degrees Celsius to theequivalent temperature ofF(c) degrees Fahrenheit. The formula is given by

F(c) =9

5c + 32, where c is a temperature in degrees Celsius.

a. What is freezing cold in degrees Celsius? in degrees Fahrenheit? Verify that the formula forFconverts correctly for freezing temperatures.

b. What is boiling hot in degrees Celsius? in degrees Fahrenheit? Verify that the formula forFconverts correctly for boiling hot temperatures.

c. Draw the graph ofFfor values ofc such that 100 c 400. What is the shape of thegraph you drew? Is this the shape of the whole graph?

d. Verify that ifc = 25, then F(c) = 77 is true. What is the contrapositive of this statement?How do you know that the contrapositive is true without additional verification?

e. What is the converse of ifc = 25, then F(c) = 77? How can you use the formula forFtoverify that the converse is true? What is the contrapositive of the converse? How do youknow that this last statement is true without additional verification?

f. There is a statement that combines a statement and its converse; its called abiconditional.

Definition: Ifp and qare statements, then the statement p if and only ifq is called abiconditional statementand is logically equivalent to ifq, thenp andifp, then q.

Write three true biconditional statements about values of the function F. Explain how youknow that the statements are true.


36/61





Unit 1: Page36 of61

Sequences as Functions Learning Task

In your previous study of functions, you have seen examples that begin with a problem situation.

In some of these situations, you needed to extend or generalize a pattern. In the followingactivities, you will explore patterns as sequences and view sequences as functions.

A sequence is an ordered list of numbers, pictures, letters, geometric figures, or just about anyobject you like. Each number, figure, or object is called a term in the sequence. For convenience,the terms of sequences are often separated by commas. In Mathematics I, we focus primarily onsequences of numbers and often use geometric figures and diagrams as illustrations and contextsfor investigating various number sequences.

1. Some sequences follow predictable patterns, though the pattern might not be immediatelyapparent. Other sequences have no pattern at all. Explain, when possible, patterns in the

following sequences:

a. 5, 4, 3, 2, 1

b. 3, 5, 1, 2, 4

c. 2, 4, 3, 5, 1, 5, 1, 5, 1, 5, 1, 7

d. S, M, T, W, T, F, S

e. 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31

f. 1, 2, 3, 4, 5, , 999, 1000

g. 1, -1, 1, -1, 1, -1,

h. 4, 7, 10, 13, 16, i. 10, 100, 1000, 10000, 100000,

The first six sequences above arefinite sequences, because they contain a finite number of terms.

The last three are infinite sequences because they contain an infinite number of terms. The threedots, called ellipses, indicate that some of the terms are missing. Ellipses are necessary for infinitesequences, but ellipses are also used for large finite sequences. The sixth example consists of thecounting numbers from 1 to 1000; using ellipses allows us to indicate the sequence without havingto write all of the one thousand numbers in it.


37/61





Unit 1: Page37 of61

2. In looking for patterns in sequences it is useful to look for a pattern in how each term relates tothe previous term. If there is a consistent pattern in how each term relates to the previous one,it is convenient to express this pattern using arecursive definition for the sequence. A

recursive definition gives the first term and a formula for how the nth

term relates to the (n1)

th

term. For each sequence below, match the sequence with a recursive definition that correctlyrelates each term to the previous term, and then fill in the blank in the recursive definition.

a. 1, 2, 3, 4, 5, I. t1 = _____ , tn = tn 1 + 5

b. 5, 10, 15, 20, 25, II. t1 = _____ , tn = tn 1 + 2

c. 1.2, 11.3, 21.4, 31.5, 41.6, 51.7, III. t1 = _____ , tn = tn 1 + 1

d. 3 2 , 5 2 , 7 2 , 9 2 , 11 2 , IV. t1 = _____ , tn = (-2) tn 1

e. 1, 2,4, 8,16, V. t1 = _____ , tn = tn 1 + 10.1

3. Some recursive definitions are more complex than those given in item 2. A recursivedefinition can give two terms at the beginning of the sequence and then provide a formula forthe nth term as an expression involving the two preceding terms, n1 and n2. It can givethree terms at the beginning of the sequence and then provide a formula for the nth term as anexpression involving the three preceding terms, n1, n2, and n3; and so forth. Thesequence of Fibonacci numbers, 1, 1, 2, 3, 5, 8, 13, 21, , is a well known sequence with such

a recursive definition.

a. What is the recursive definition for the Fibonacci sequence?

b. The French mathematician Edouard Lucas discovered a related sequence that has manyinteresting relationships to the Fibonacci sequence. The sequence is 2, 1, 3, 4, 7, 11, 18,29, , and it is called the Lucas sequence. For important reasons studied in advancedmathematics, the definition of the Lucas numbers starts with the 0th term. Examine thesequence and complete the recursive definition below.

c. What is the 8th term of the Lucas sequence, that is, the term corresponding to n = 8?

d. What is the 15th

term?


38/61





Unit 1: Page38 of61

n = 1 n = 2 n = 3 n = 4 n = 5

4. Recursive formulas for sequences have many advantages, but they have one disadvantage. If

you need to know the 100

th

term, for example, you first need to find all the terms before it. Analternate way to define a sequence uses aclosed form definition that indicates how todetermine the nth term directly, without the need to calculate other terms. Match eachsequence below with a definition in closed form. Verify that the closed form definitions youchoose agree with the five terms given for each sequence.

a. 5, 10, 15, 20, 25, I. 3, 1, 2, 3, ..nt n for n

b. 1, 0, 2, 0, 3, 0, 4, II. 5 , 1, 2, 3, ..nt n for n

c. 5, 9, 13, 17, 21, 25, III. 4 1, 1, 2, 3, ..nt n for n

d. 1, 8, 27, 64, 125, . . . IV.

1, if =1, 3, 5, ...

2

0, if = 2, 4, 6, ...

n

nn

t

n

Sequences do not need to be specified by formulas; in fact, some sequences are impossible tospecify with formulas. When sequences are given by formulas, the two types of formulas mostcommonly used are the ones explained above: recursive and closed form. In item 4, you probablynoticed that a closed form definition for a sequence looks very much like a formula for a function.If you think of the index values as inputs and the terms of the sequence as outputs, then anysequence can be considered to be a function, whether or not you have a formula for the n-th termof the sequence.

5. Consider the sequence of dot figures below.

a. The sequence continues so that the number of dots in each figure is the next squarenumber. Why are these numbers called square numbers?

b. What is the next figure in the sequence? What is the next square number?

c. What is the 25th

square number? Explain.


39/61





Unit 1: Page39 of61

d. Is 156 a square number? Explain.

e. Let g be the function that gives the number of dots in the nth

figure above. Write a formulafor this function.

f. Make a table and graph for the function g.

g.

What is the domain of the function g?

h. What is the range of the function g?

6. Consider a functionA that gives the area of a square of side-length s, shown below (and to theright).

a. What is the area of a square of side length 6?

b. Find a formula for the functionA.

c. Make a table and graph for the functionA.

d. Use your formula to find the area of squares with

the following side lengths: 7, , 4.2, 3 , and 200.

e. What is the domain ofA (in the given context)?

f. What are the limiting case(s) in this context?Explain.


40/61





Unit 1: Page40 of61

g. Change the graph to extend the domain ofA to include the limiting case(s). Whatchange did you make in the graph?

7. Compare the functions g andA from items 5 and 6.

a. Explain how similarities and differences in the formulas, tables, and graphs arise fromsimilarities and differences in the contexts.

b. Are these functions equal? Explain.

8. Look back at the finite sequences in 1a and 1b above. Consider the sequence in item1, part a,to be a function named h and the sequence in item 1, part b to be a function named k.

a. Make tables and graphs for the functions h and k.

b. How many points are included in each graph?

c. Determine the following:

domain of h __________ range of h __________domain of k__________ range of k__________

d. Are these functions equal? Explain.

e. Write a formula for that gives h(n) for each number n in the domain ofh. What kind offunction did you write?

f. Can you write a linear function formula for the function k? Explain why or why not?

g. Consider the linear function dgiven by the formulad(x) = 10x, for each real numberx.

Make a table and graph for the function d.

h. Compare the tables, graphs, and formulas for the functions h and d. In your comparison,pay attention to domains, ranges, and the shapes of the graphs.


41/61





Unit 1: Page41 of61

Walking, Falling, and Making Money

In previous mathematics courses, you studied the formula distance = ratex time, which is usuallyabbreviated d rt. If you and your family take a trip and spend 4 hours driving 200 miles, thenyou can substitute 200 for d, 4 for t, and solve the equation 200 = r*4 to find that r= 50. Thus, wesay that the average speed for the trip was 50 miles per hour. In this task, we develop the idea ofaverage rate of change of a function, and see that it corresponds to average speed in certainsituations.

1. To begin a class discussion of speed, Dwain and Beth want to stage a walking race down theschool hallway. After some experimentation with a stop watch, and using the fact that theflooring tiles measure 1 foot by 1 foot, they decide that the distance of the race should be 40

feet and that they will need about 10 seconds to go 40 feet at a walking pace. They decide thatthe race should end in a tie, so that it will be exciting to watch, and finally they make a tableshowing how their positions will vary over time. Your job is to help Dwain and Beth makesure that they know how they should walk in order to match their plans as closely as possible.

Time (sec.) 0 1 2 3 4 5 6 7 8 9 10

Dwains position (ft.) 0 4 8 12 16 20 24 28 32 36 40

Beths position (ft.) 0 1 3 6 10 15 20 25 30 35 40

a. Draw a graph for this data. Should you connect the dots? Explain.

b. How can you tell that the race is supposed to end in a tie? Provide two explanations.

c. Who is ahead 5 seconds into the race? Provide two explanations.

d. Describe how Dwain should walk in order to match his data. In particular, should Dwainsspeed be constant or changing? Explain how you know, using observations from both thegraph and the table.


42/61





Unit 1: Page42 of61

e. Describe how Beth should walk in order to match his data. In particular, should Bethsspeed be constant or changing? Explain how you know, using observations from both thegraph and the table.

f. In your answers above, sometimes you paid attention to the actual data in table. At othertimes, you looked at how the data change, which involved computing differences betweenvalues in the table. Give examples of each. How can you use the graph to distinguishbetween actual values of the data and differences between data values?

g. Someone asks, What is Beths speed during the race? Kellee says that this question does

not have a specific numeric answer. Explain what she means.

h. Chris says that Beth went 40 feet in 10 seconds, so Beths speed is 4 feet per second. ButKellee thinks that it would be better to say that Beths average speedis 4 feet per second.Is Chriss calculation sensible? What does Kellee mean?

i. Taylor explains that to compute average speed over some time interval, you divide thedistance during the time interval by the amount of time. Compute Dwains average speedover several time intervals (e.g., from 2 to 5 seconds; from 3 to 8 seconds). What do younotice? Explain the result.

j. What can you say about Beths speed during the first five seconds of the race? What aboutthe last five seconds? Explain.


43/61





Unit 1: Page43 of61

k. Trey wants to race alongside Dwain and Beth. He wants to travel at a constant speedduring the first five seconds of the race so that he will be tied with Beth after five seconds.At what speed should he walk? Explain how Treys walking can provide an interpretation

of Beths average speed during the first five seconds.

In describing relationships between two variables, it is often useful to talk about the rate of changeof one variable with respect to the other. When the rate of change is not constant, we often talkabout average rates of change. If the variables are calledx andy, andy is the dependent variable,then

Average rate of change ofy with respect tox =x

y

inchangethe

inchangethe.

Wheny is distance andx is time, the average rate of change can be interpreted as an average speed,

as we have seen above.

2. Earlier you investigated the distance fallen by a ball dropped from a high place, such as theTower of Pisa. In that problem,y, measured in meters, is the distance the ball has fallen andx,measured in seconds, is the time since the ball was dropped. You saw thaty is a function ofx,and the relationship can be approximated by the formulay =f(x) = 5x

2. You completed a table

like the one below.

x 0 1 2 3 4 5 6 x

2 0 1 4 9

y =f(x) = 5x2 0 5 20

a. What is the average rate of change of the height of the ball with respect to time during thefirst second after the ball is dropped? In what units is the average rate of changemeasured?

b. What is the average rate of change of the height of the ball during the second second afterthe ball is dropped? after the third, fourth, and fifth? What do these answers say about tothe speed of the ball?


44/61





Unit 1: Page44 of61

c. Find the average rate of change over the first five seconds. Is this the same average speedas the average speed for the fifth second during which the ball is falling? Explain.

d. What is the average rate of change over the first two seconds?

e. What is the average rate of change from 2 to 5 seconds after the ball is dropped?

f. What is the average rate of change from t= 1.5 to t= 2?

g. Why is it important to use the phrase average rate of change or average speed for yourcalculations in this problem?

Average speed is a common application of the concept of average rate of change but certainly not

the only one. There are many applications to analyzing the money a company can make fromproducing and selling products. The rest of t

9-12 math i student edition unit 1 function families

Documents